As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction. An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves. We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks. The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic. We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.

中文翻译：

---

通过粘合向量束构建共界

正如 Gabriel 已经观察到的，通过粘合仿射开子集获得的方案上的相干滑轮可以通过简单的粘合构造来描述。Ferrand 的一个例子表明，这对于沿封闭浸入的推出通常是失败的，尽管胶合结构仍然适用于平坦的连贯滑轮。我们表明，通过进一步将这种粘合构造限制为向量丛，我们可以沿着称为 Adams 堆栈的某些代数堆栈的任意态射（以及更一般的共限）构造推出。这个事实的证明使用了广义的 Tannaka 对偶性和 Deligne 论证的变体，以证明存在以任意特性工作的纤维函子。我们将这个版本的 Deligne 存在定理用于纤维函子，作为识别具有地图集的堆栈的新方法。